There is a need for scientists and engineers to have a numerical library
that:

is free (in the sense of freedom, not in the sense of gratis; see the
GNU General Public License), so that people can use that library,
redistribute it, modify it ...

is written in C using modern coding conventions, calling conventions,
scoping ...

is clearly and pedagogically documented; preferably with TeXinfo, so as
to allow online info, WWW and TeX output.

uses top quality state-of-the-art algorithms.

is portable and configurable using autoconf and automake.

basically, is GNUlitically correct.

There are strengths and weaknesses with existing libraries:

Netlib (http://www.netlib.org/) is probably the most advanced set
of numerical algorithms available on the net, maintained by AT&T.
Unfortunately most of the software is written in Fortran, with strange
calling conventions in many places. It is also not very well collected,
so it is a lot of work to get started with netlib.

GAMS (http://gams.nist.gov/) is an extremely well organized set
of pointers to scientific software, but like netlib, the individual
routines vary in their quality and their level of documentation.

Numerical Recipes (http://www.nr.com,
http://cfata2.harvard.edu/nr/) is an excellent book: it explains the
algorithms in a very clear way. Unfortunately the authors released the
source code under a license which allows you to use it, but prevents you
from re-distributing it. Thus Numerical Recipes is not free in
the sense of freedom. On top of that, the implementation suffers
from fortranitis and other
limitations. [http://www.lysator.liu.se/c/num-recipes-in-c.html]

SLATEC is a large public domain collection of numerical routines
in Fortran written under a Department of Energy program in the
1970's. The routines are well tested and have a reasonable overall
design (given the limitations of that era). GSL should aim to be a
modern version of SLATEC.

NAG and IMSL both sell high-quality libraries which are
proprietary. The NAG library is more advanced and has wider scope than
IMSL. The IMSL library leans more towards ease-of-use and makes
extensive use of variable length argument lists to emulate "default
arguments".

Numerical Algorithms with C G. Engeln-Mullges, F. Uhlig. A nice
numerical library written in ANSI C with an accompanying
textbook. Source code is available but the library is not free software.

NUMAL A C version of the NUMAL library has been written by
H.T. Lau and is published as a book and disk with the title "A Numerical
Library in C for Scientists and Engineers". Source code is available but
the library is not free software.

C Mathematical Function Handbook by Louis Baker. A library of
function approximations and methods corresponding to those in the
"Handbook of Mathematical Functions" by Abramowitz and Stegun. Source
code is available but the library is not free software.

CCMATH by Daniel A. Atkinson. A C numerical library covering
similar areas to GSL. The code is quite terse. Earlier versions were
under the GPL but unfortunately it has changed to the LGPL in recent
versions.

CEPHES A useful collection of high-quality special functions
written in C. Not GPL'ed.

WNLIB A small collection of numerical routines written in C by
Will Naylor. Public domain.

MESHACH A comprehensive matrix-vector linear algebra library
written in C. Freely available but not GPL'ed (non-commercial license).

[JT section: written by James Theiler, and not incorporated into the
above motivation]

And we furthermore promise to try as hard as possible to document
the software: this will ideally involve discussion of why you might want
to use it, what precisely it does, how precisely to invoke it,
how more-or-less it works, and where we learned about the algorithm,
and (unless we wrote it from scratch) where we got the code.
We do not plan to write this entire package from scratch, but to cannibalize
existing mathematical freeware, just as we expect our own software to
be cannibalized.

The long-term goal will be to provide a framework to which the real
numerical experts (or their graduate students) will contribute. Such
contributors will have to contribute copylefted software and they cannot
mind that evil profit-making companies (such as those which sell
"environments") might use this software as part of their packages.